formal 1 Languages – 1 1. Introduction.......................................................................................................................................2 2. The Basic Ingredients of a Formal Language ....................................................................................2 3. A Simple Example: The Language of Sentential Logic.....................................................................3 4. Another Simple Example of a Formal Language...............................................................................3 5. Back to SL........................................................................................................................................4 6. Object Language versus MetaLanguage ............................................................................................5 7. Use versus Mention ..........................................................................................................................6 8. Numerals versus Numbers................................................................................................................6 9. The Single-Quote Convention...........................................................................................................7 10. Off-Set Notation ...............................................................................................................................8 11. Elaborating on the Basic Single-Quote Scheme ................................................................................8 12. Single-Quotes versus Double-Quotes...............................................................................................9 13. A Big Difference Between Single-Quotes and Double-Quotes.........................................................9 14. Another Example of Use-Mention...................................................................................................11 15. Symbolizing Meta-Linguistic Statements – 1 ..................................................................................12 16. The Role of the Tilde Symbol in English+......................................................................................14 17. A Conservative Formulation of English+ .......................................................................................15 18. A More Liberal Formulation of the Grammar of English+..............................................................16 19. Other Approaches to Symbolic Use versus Mention.......................................................................17 20. Symbolizing Meta-Linguistic Statements – 2 ..................................................................................18 21. String Addition...............................................................................................................................19 22. Quine Quotes ..................................................................................................................................20 23. A Formal Account of Quine-Quotation...........................................................................................21 24. Dropping Quine Quotes Altogether.................................................................................................22 25. Summary: Approaches to Metalinguistic Singular Terms ...............................................................23 26. Appendix: General Translation Scheme for Elementary Logic.......................................................24 27. Exercises ........................................................................................................................................26 28. Answers to Exercises .....................................................................................................................28 Formal Languages – 1 2 1. Introduction Logic provides tools for systematically appraising reasoning. Central among these tools are formal languages. The word ‘formal’ has a number of uses, at least two of which figure in logic. One use pertains to the word ‘form’, as used in contrast to ‘content’ or ‘matter’. This is related to the crucial idea, which traces to Aristotle, that whether an argument is valid is a function of its form. The other use of the word ‘formal’ is in contrast to ‘informal’. A related notion is the notion of rigor – formal languages are rigorous in the sense that they are stiff (think of the term ‘rigor mortis’) . Related to the formal/informal distinction is the distinction between natural language and artificial language; formal languages are a species of artificial language. Leaving lexicography aside for the moment, 20th Century logic offers a concept of formal that is somewhat more specific. According to this concept, a formal language is a language that is computationally recognizable (“computable” for short). In other words, for a language to be formal in this sense, it must be possible in principle to program a computer so that it can distinguish between items in that language and items not in that language. It seems fairly clear that, if a language is formal in this sense, it is also formal in the more conventional sense. 2. The Basic Ingredients of a Formal Language In order to specify a formal language o, one does two things, at a minimum. (1) one specifies the underlying set of symbols on which o is built; (2) one specifies which strings of symbols count as well-formed in o. This is often described by saying that, in specifying a formal language o, one specifies (1) the vocabulary of o (2) the rules of formation of o Note: the term ‘symbol’ may be replaced by the term ‘character’, as used in many computer languages, such as Basic, Pascal, and C++. Notice in this connection that the term ‘string’ is also used in many computer languages, usually to mean string of characters. Let us not worry too much about exactly what a symbol (character) is; for example, how exactly can we tell when we have a symbol rather than a natural number, a triangle, or a potato? Intuitively, symbols are the fundamental units of writing – the sorts of things one can write down on paper, key into a computer, inscribe in stone, spray-paint on a building, etc. You are looking at lots of examples of symbols as you read this. You are also looking at lots of examples of strings of symbols. A string of symbols is just a (finite) bunch of symbols strung together in a row [notice that line breaks are ignored]. Most natural languages are formulated in this purely linear fashion. On the other hand, mathematics provides many examples of formulas that are not linear, but two- dimensional. Generally, however, there are systematic ways to “flatten” these two-dimensional formulas. In any case, we will concentrate on linear (flat) languages. Some strings of symbols are grammatically significant (well-formed), and some are not. The text you are reading is full of examples of both. Pick two locations in this text; the literal material lying Formal Languages – 1 3 between these two points is a string of symbols, but it need not (in isolation) be a grammatically significant unit; it need not be well-formed. In summary (where o is a formal language): The vocabulary of o is the set of symbols employed by o. The rules of formation of o specify which strings of symbols count as well- formed. 3. A Simple Example: The Language of Sentential Logic The student of elementary logic is already familiar with a number of formal languages, including the language of sentential logic (SL). The formal presentation of the syntax of this language, oS, may be given as follows. The language of SL – o S: Vocabulary: (1) upper case Roman letters: A B C …… (2) special symbols: ; & Ú ® « ( ) (3) nothing else. Rules of Formation: (1) every upper case Roman letter is a formula; (2) if Æ is a formula, then so is: ;Æ (3) if Æ1 and Æ2 are formulas, then so are: (Æ1 & Æ2) (Æ1 Ú Æ2) (Æ1 ® Æ2) (Æ1 « Æ2) (4) nothing else is a formula. Here, the block letter ‘Æ’ is used in order to suggest ‘string’. The role of funny-looking letters will be explained shortly. 4. Another Simple Example of a Formal Language In order to emphasize that the term ‘language’ in ‘formal language’ should not be taken too literally, we consider another example of a formal language, which we call o2. Vocabulary: Formal Languages – 1 4 (1) upper case Roman letters: A B C …… (2) nothing else is a symbol of o2. Rules of Formation: (1) if a string Æ begins with ‘P’, then it is a formula. (2) nothing else is a formula of o2. Admittedly, o2 is a silly formal language, with no obvious use in logic. Nevertheless, it is a formal language by our official criteria. 5. Back to SL Recall that, as it is officially presented in Section 4, there are exactly 26 simple (atomic) formulas in the language of SL. Although this particular formal language is entirely adequate for all sensible exercises in elementary logic, it is not entirely adequate for all theoretical uses of sentential logic. The problem is that there are not enough atomic formulas. The usual informal way to rectify this problem is to add numerical subscripts to the letters, so as to yield infinitely many atomic formulas. So in addition to ‘P’, ‘Q’, etc., we have ‘P1’, ‘Q2’, etc. This produces a new theoretical problem, however – how to formalize our rules of formation so that we can program our hypothetical computer to recognize formulas in the expanded language. An alternative technique of generating infinitely many atomic formulas proceeds as follows. Vocabulary: (1) upper case Roman letters: A B